The Derivative-Instantaneous rate of change

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The Derivative-Instantaneous rate of change
The derivative of a function, f at a specific
value of x, say a is a value given by
The derivative of a function, f as a function of
x, is called f ?(x) and is given by
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Find the derivative of
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Related problems
1) Find the slope of f (x) at x 3, x -2
2) Write the equation of the tangent line at x
-2
3) Find the point on f (x) for which the slope is
2
4) Find the point for which f (x) has a
horizontal tangent line
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Solutions
1) Find the slope of f (x) at x 3, x -2
so
2) Write the equation of the tangent line at x
-2
use the point-slope formula
Find the value of y
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Solutions
3) Find the point on f (x) for which the slope is
2
The point is (1, 6)
4) Find the point for which f (x) has a
horizontal tangent line
The point is (2/3, 17/3)
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Derivatives at Endpoints are one-sided limits.
Figure 2.7 Derivatives at endpoints are
one-sided limits.
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How a derivative can fail to exist
Which of the three examples are the
functions continuous?
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The graph of a function
Where f(x) is increasing (slope is positive)
Horizontal tangent (slope 0)
Where f(x) is decreasing (Slope is negative)
The graph of the derivative (slope) of the
function
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3.3 Differentiation formulas
Constant
Simple Power rule
Constant multiple rule
Sum and difference rule
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Find the derivative function for
rewrite
rewrite
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Rules for Finding Derivativesu and v are
functions of x.
Simple Power rule
Constant multiple rule
Sum and difference rule
Product rule
Quotient rule
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Differentiate
Product rule
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Differentiate
Quotient rule
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Find the derivative function for
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Velocity. The particle is moving forward for the
first 3 seconds and backwards the next 2 sec,
stands still for a second and then moves forward.
forward motion means velocity is positive
backward motion means velocity is negative
If velocity 0, object is standing still.
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3.4 applications
A dynamite blast blows a heavy rock straight up
with a launch velocity of 160 ft/sec. Its height
is given by s -16t2 160t.
a) How high does the rock go?
b) What is the velocity when the rock is 256 ft.
above the ground on the way up? On the way down?
c) What is the acceleration of the rock at any
time?
d) When does the rock hit the ground? At what
velocity?
The graphs of s and v as functions of time s is
largest when v ds/dt 0. The graph of s is not
the path of the rock It is a plot of height
versus time. The slope of the plot is the rocks
velocity graphed here as a straight line.
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A dynamite blast blows a heavy rock straight up
with a launch velocity of 160 ft/sec. Its height
is given by s -16t2 160t.
a) How high does the rock go?
Maximum height occurs when v 0.
s -16t2 160t
v s -32t 160
-32t 160 0
t 5 sec.
At t 5, s -16(5)2 160(5) 400 feet.
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A dynamite blast blows a heavy rock straight up
with a launch velocity of 160 ft/sec. Its height
is given by s -16t2 160t.
b) What is the velocity when the rock is 256 ft.
above the ground on the way up? On the way down?
-16t2 160t 256 -16t2 160t 2560 -16(t2 - 10t
16)0 -16(t 2) (t- 8) 0 t 2 or t 8
Set position 256
Find the times
v -32t 160 at t 2 v-32(2)160 96
ft/sec. at t 8 v-32(8)160 -96 ft/sec
Substitute the times into the velocity function
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A dynamite blast blows a heavy rock straight up
with a launch velocity of 160 ft/sec. Its height
is given by s -16t2 160t.
c) What is the acceleration of the rock at any
time?
s -16t2 160t
v s -32t 160
a v s -32ft/sec2
d) When does the rock hit the ground? At what
velocity?
s -16t2 160t 0 t 0 and t 10
Set position 0
v -32t 160 v -32(10) 160 -160 ft/sec.
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3.5 Derivatives of trig functions-formulas needed
sin(xh) sin xcos hcos xsin h
cos(xh) cos xcos h- sin xsin h
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Derivative of y sin x
0 cos(x)1 cos (x)
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3.5 Derivatives of Trigonometric Functions
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Slope of y cos x
Figure 25 The curve y sin x as the graph
of the slopes of the tangents to the curve y
cos x.
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Find the derivatives